Business Statistics - I · Diagrammatic Presentation: One Dimensional and Two Dimensional Diagrams...

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Business Statistics - I(As per the New CBCS Syllabus of 2nd Year, 3rd Semester, B.Com.

(All Streams) of All the Universities in Telangana State w.e.f. 2016-17)

B.M. AGGARWALB.Sc. (Hons), M.A. (Maths),

Grad. IETE (FIETE),Certificate Courses in Microwave & Satellite Engineering,

Visiting Professorand

Author of Booksin the Area of Mathematics, Statistics and Operations Research.

Published by : Mrs. Meena Pandey for Himalaya Publishing House Pvt. Ltd.,Ramdoot, Dr. Bhalerao Marg, Girgaon, Mumbai - 400 004Phone: 022-23860170/23863863; Fax: 022-23877178E-mail: [email protected]; Website: www.himpub.com

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© AUTHORNo part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or byany means, electronic, mechanical, photocopying, recording and/or otherwise without the prior written permissionof the author and the publisher.

First Edition : 2017Second Revised Edition : 2018

Dedicatedwith extreme reverence to

my fatherwho taught me

some concepts of mathematicswhich have proved to mean asset and inheritance.

Mohan

PREFACE

The book has been written strictly as per the syllabus prescribed by theTeleganna University for their B.Com., Third Semester.

The salient features of the book are:

1. The subject matter has been dealt with exhaustively in a limited spacewithout compromising with the needs of the students.

2. Complete solutions to unsolved exercises have been given to help thestudents and the teachers as well.

3. MCQs, short answer questions and fill in the blanks have been appendedat the end of each chapter to give conceptual understanding to the readers.

4. A large number of questions selected from the question papers of differentuniversities have been given as solved examples to apprise the studentsabout the latest trends.

I am thankful to my wife Mrs Prem Lata Jain who spares enough time forme to complete my assignments.

I am indebted to M/s Himalaya Publishing House Pvt. Ltd. and their staffspecially Ms. Nimisha and Ms. Lalita for their utmost co-operation in completingthe project.

In the end, I request the readers to send healthy suggestions for the improvementof the book.

B.M. Aggarwal

SYLLABUS

Max. Marks: 100 Exam Duration: 3 Hours

Objective: To inculcate analytical and computational ability among the students.

UNIT-I: INTRODUCTIONOrigin and Development of Statistics – Definition – Importance and Scope – Limitations

of Statistics – Distrust of Statistics.Statistical Investigation: Planning of Statistical Investigation – Census and Sampling Methods

– Collection of Primary and Secondary Data – Statistical Errors and Approximation – Classificationand Tabulation of Data – Frequency Distribution.

UNIT-II: DIAGRAMMATIC AND GRAPHIC PRESENTATIONDiagrammatic Presentation: One Dimensional and Two Dimensional Diagrams – Pictograms

– Cartograms Graphic Presentation: Technique of Construction of Graphs – Graphs of FrequencyDistribution – Graphs of Time Series or Histograms.

UNIT-III: MEASURES OF CENTRAL TENDENCYIntroduction – Significance – Arithmetic Mean – Geometric Mean – Harmonic Mean –

Mode – Median – Quartiles and Percentiles – Simple and Weighted Averages – Uses and Limitationsof Different Averages.

UNIT-IV: MEASURES OF DISPERSION, SKEWNESS AND KURTOSISMeasures of Dispersion: Significance – Characteristics – Absolute and Relative Measures

– Range – Quartile Deviation – Mean Deviation – Standard Deviation – Coefficient of Variation.

Measures of Skewness – Karl Pearson’s Coefficient of Skewness – Bowley’s Coefficientof Skewness – Kelly’s Measure of Skewness – Kurtosis: Mesokurtosis, Platykurtosis and Leptokurtosis.

UNIT-V: CORRELATIONMeaning – Types – Correlation and Causation – Methods: Scatter Diagram – Karl Person’s

Coefficient of Correlation – Probable Error and Interpretation of Coefficient of Correlation –Rank Correlation – Concurrent Deviation Method.

1. Introduction to Statistics 1 – 18

2. Statistical Investigation and Planning 19 – 48

3. Presentation of Statistical Data 49 – 116

4. Measures of Central Tendency (Statistical Averages) 117 – 183

5. Measures of Dispersion 184 – 218

6. Skewness and Kurtosis 219 – 231

7. Correlation 232 – 266

University Question Papers 267 – 270

Solved University Question Papers 271 – 283

CONTENTS

1

1

INTRODUCTION TOSTATISTICS

1.1 INTRODUCTION AND ORIGIN OF STATISTICSThe collection of numerical data on various aspects of life is almost as old as civilization itself. Prof.

P.C. Mahalanobis said, “From time immemorial, men must have been computing information for peace andwar. Statistics in this sense is as old as statecraft.”

Initially, the concept of statistics started with data collection and something like ‘Political arithmetic’.The kings used to collect the information about the people in their states, i.e., they used to collect the dataregarding conditions under which people live and work, their health, their incomes and wealth. So, thissubject developed as ‘a King’s subject’ or as ‘a science of Kings’. Kings also used to collect data such aspopulation, manpower, wealth in the form of land, buildings and other assets, to know the condition of theirstates. R.A. Fisher regards the science of statistics as “Mathematics applied to observational data.”S.S. Wilks in 1951 remarked, “In a democracy, the citizen lives in a world of facts and figures. He makesdecisions all of the time on the basis of large or small amounts of information. He carries on in a massproduction economy .......” Perhaps, H.G. Wells was right when he said, “Statistical thinking will one daybe as necessary for efficient citizenship as the ability to read and write.” Statistics is not a science but ascientific method and scientific methods are essential for better efficiency. Nowadays, practically, there is nosphere of life where statistics do not find applications. In the words of M. Walker, “Today, to a very strikingdegree, our culture has become a statistical culture. Even a person who may never have heard of an indexnumber is affected in an intimate fashion by the gyrations of those index numbers which describe the costof living. Even on the most elementary level, it is impossible to understand psychology, sociology, economics,finance or the physical sciences without some general idea of the meaning of an average, of variation, ofcorrelation, of sampling, of how to interpret charts and tables. The State legislatures deal continuallywith matters in which it is impossible to reach a sound decision without weighing statistical evidence.”

The word ‘statistics’ appears to have been derived from the Latin word ‘status’, Italian word ‘statista’,German word ‘Statistik’ or French word ‘statistique’ all referring to the political State. Achenwall called the

2 BUSINESS STATISTICS - I

father of statistics, in 1749 defined statistics as “The political science of the several centuries”. In the earlyyears, ‘statistics’ connoted a collection of facts about the State or the people in the State for administrativeor political purposes. Thus, it helped in conducting the census operations and collecting data about thesocial and economic conditions of different groups of persons living in different parts of the country.

The word ‘statistics’, until the last quarter of the 19th century, was used to signify not only numbersand quantitative information but also facts calculated to illustrate the conditions and prospects of thesociety. But, by the turn of the 20th century, the term ‘statistics’ became identified with quantitative informationand today this is almost the exclusive emphasis which is given to the subject.

Lowell J. Reed in 1952 said, “although statistics is in its infancy in a scientific sense, it is certain toinfluence profoundly all future scientific thinking.”

1.2 EMPIRICAL DATA AND QUANTITATIVE MEASUREMENTSAn empirical investigation consists of collecting information through observations and analysing the

information for drawing conclusions. When the information or observations are recorded in quantity, theinformation is said to have been quantified. For instance, weights, heights, age, number of accidents,production, sales, etc., are phenomena which are measurable in quantity along a scale.

The systematic and scientific treatment of quantitative measurements* is precisely known as statistics.The term statistics covers all fields of study wherein quantitative measurement of observations is possibleand need analysis.

1.3 DEFINITIONS OF STATISTICSThe word statistics is used in two different but interrelated ways:(i) As a plural noun (ii) As a singular noun.As a plural noun, statistics refers to statistical data. As a singular noun, it refers to statistical methods.

Statistical methods include collection, classification, presentation, analysis and interpretation of data.Therefore, on the whole, statistics refers to statistical data and the method of obtaining and analysing datain order to base decisions on them.

In this sense, the most comprehensive and exhaustive definition has been given by Prof. HoraceSecrist. He defines: “By statistics, we mean aggregate of facts affected to a marked extent by multiplicity ofcauses, numerically expressed, enumerated or estimated according to reasonable standards of accuracy,collected in a systematic manner for a predetermined purpose and placed in relation to each other.”

According to this definition, statistics in plural sense should have the following characteristics:(i) They are aggregate of facts.

(ii) They are affected to a marked extent by multiplicity of causes.(iii) They are numerically expressed.(iv) They are enumerated or estimated according to a reasonable standard of accuracy.(v) They are collected in a systematic manner.

(vi) They are collected for a predetermined purpose.(vii) They are placed in relation to each other.

* All data are not statistical. Data in quantitative form [numerical form] only can be a statistical data, but all numericaldata also are not statistical (Isolated facts expressed numerically do not constitute data. For example, the wage earnedby an industrial worker at anyone time, taken by itself, is not a statistical data but taken as a part of a mass ofinformation, may be a statistical data). Thus, quantitative data are statistical when they relate measurement of masses,and not statistical when they relate an individual item or event as a separate entity.

INTRODUCTION TO STATISTICS 3

(i) They are aggregate of facts. Statistics are concerned with ‘population’ of facts rather than with asingle fact. Single or unrelated facts do not throw light on any problem. In short, “not a datum** but the dataare the subject matter of statistics.” Suppose a student ‘X’ is said to be poor in mathematics. It is not a caseof statistics but if the marks of all the students in a class are recorded, then it becomes statistics. No doubt,all statistics are expressed in numbers but all numbers are not statistics.

(ii) They are affected to a marked extent by multiplicity of causes. Statistics refer to measurement ofphenomena in a complex universe. They are related to other measurements. They grow out of a variety ofcircumstances differing among themselves, and are constantly subject to change. They are not traceable toa single cause.

(iii) They are numerically expressed. For example, rise in prices may be due to many factors like shortsupply, untimely rainfall, less production, increase in demand, shortage of power, rise in wages of labourers,etc.

In the words of Prof. H. Secrist, “The statistical approach to a subject is numerical. Things, attributesand conditions are counted, totalled, divided, subdivided and analysed.” This means that the facts, to becalled statistics, must be numerically expressed. Qualitative expressions like good, poor, young, old, etc., donot form part of statistical studies unless expressed numerically in some form.

For example, saying Aishwarya Rai is Miss World is not a statistics but saying that in 1994 AishwaryaRai was selected as Miss World is a statistics.

(iv) They are enumerated or estimated according to a reasonable standard of accuracy. Thischaracteristic of ‘statistics’ is related with the collection of data. Data can be collected either by enumerationor by estimation. If the sphere of enquiry is limited, then enumeration is resorted to. The data collected byenumeration are more correct. When the sphere of enquiry is wide, enumeration becomes a difficult job. Insuch cases, estimation is the only way out. According to Dr. A.L. Bowley, “Great numbers are not countedcorrectly to a unit, they are estimated.” It is natural that the estimated figures cannot be absolutely accurate.Prof. Secrist himself has stated that, “what are reasonable standards depend upon the purpose which thestatistics are to serve.” Prof. Secrist has further said, “No absolute criterion can be established for allcases, where precision is required, accuracy is necessary, where general impressions are sufficient,appreciable error may be tolerated.”

(v) They are collected in a systematic manner. The collection of facts should be systematic. Haphazardlycollected figures are not desirable because they may lead to erroneous conclusions. According to Prof.Secrist, “Stray and loose bits of quantitative information, hearsay and unrelated material gleaned hereand there, from indiscriminate sources having no common basis of selection, even when numerical can betermed statistics only by confusion of terms.”

(vi) They are collected for a predetermined purpose. Statistics should be collected for a specific, well-defined and predetermined purpose. This will facilitate the collection of requisite information. For example, ifwe are to collect statistics of agricultural production, we must decide beforehand, the regions, commoditiesand the period (or periods) for which data are required.

(vii) They are placed in relation to each other. Statistics are collected for making comparisons betweenrelated phenomena over a specific region or period. For example, the population figures of India, U.K.,U.S.A., China, etc. can be compared only if they belong to the same census year. Thus, for a valid comparison,statistics are required to be put in relation to each other. It would be meaningless to compare the weight ofhorses with those of human beings.

* * Datum means a piece of information.


In the absence of the above conditions, numerical expressions of facts cannot be treated as ‘statistics’.Therefore, one can say “all statistics are numerical data but all numerical data are not statistics.” Forexample, logarithms are numbers but they cannot be called as statistics because they do not conform to theabove stated characteristics.

The other important definitions, defining statistics in the plural sense are:(i) Webster defines statistics as, “Statistics are classified facts representing the condition of the

people in a state.......especially those facts which can be stated in numbers or in tables ofnumbers or in any tabular or classified arrangement.” This definition limits the scope of statistics.According to this definition, only those facts are called statistics which are related to the conditionof the people in a State. Facts regarding Physics, Chemistry, Sociology and Psychology will notbe called statistics. In the present age, statistics are collected in respect of all aspects of humanactivity. Hence, this definition is inadequate and incomprehensive.

(ii) According to Dr. A.L. Bowley, “Statistics are numerical statements of facts in any department ofenquiry, placed in relation to each other.” According to this definition, statistics have threecharacteristics, viz., (a) they are numerical statements of facts, (b) they are concerned with an enquiry,and (c) they are placed in relation to each other for comparison. Other characteristics are not includedin this definition. This definition does not speak about different statistical methods also.

(iii) Yule and Kendall state that, “By Statistics, we mean quantitative data affected to a marked extentby a multiplicity of causes.” This definition is also incomplete as it mentions only two characteristicsof statistics, viz., (a) they are quantitative, and (b) they are affected by multiplicity of causes.

(iv) According to Connor, “Statistics are measurements, enumerations or estimates of natural orsocial phenomena, systematically arranged so as to exhibit their interrelations.” This definitionstates that (a) statistics are measurements, estimates or enumerations, (b) they relate to somenatural or social phenomena and (c) they are systematically arranged to exhibit their interrelations.

(v) According to Tuttle, “Statistics are measurements, enumerations or estimates of natural andsocial phenomena, usually systematically arranged, analysed and presented so as to exhibitimportant relationships among them.” This definition also does not include all the characteristicsof statistics.

1.4 STATISTICS AS A SINGULAR NOUN, i.e., STATISTICS AS STATISTICALMETHODSDr. Bowley has said, “It will be better to define Statistics as a method.” Based on this concept,

statistics is what statistics does or what statisticians do. In statistical methods, the word ‘statistics’ hasbeen treated in singular form or a body of knowledge in the area of applied mathematics.

Some of the definitions of statistics under this concept are:1. Incomplete and Traditional Definitions. Dr. Bowley has given a number of incomplete definitions

which touch only some of the aspects of statistics.(i) “Statistics may be called the science of counting.” He himself further observes, “Great

numbers are not counted..... they are estimated.”(ii) “Statistics may rightly be called the science of averages.”

(iii) “Statistics is the device for abbreviating and classifying the statements and making clearthe relations.”

(iv) “Statistics is the science of measurement of social organism regarded as a whole in all itsmanifestations.” This definition limits the application of the statistical methods in social


sciences only. Dr. Bowley himself at another place expresses contrary view that, “Statistics isnot merely a branch of political economy nor it is confined to anyone science. Statistics canbe applied in varied fields.” W.I. King while criticising this definition observes that, “Thisstatement limits the science to only one field–that of man and his activities. Modern Statisticstakes into consideration, however, biological, astronomical and physical as well as socialphenomena and hence the definitions obviously are too narrow.”

2. Like Dr. Bowley, Prof. Boddington defines statistics as, “Statistics is the science of estimates andprobabilities.” This definition is also not complete.

3. “The subject ‘Statistics’ is concerned with the collection, presentation, description and analysisof data which are measurable in numerical terms.” –P.H. Karmel

4. “Statistics deals with the collection, classification and tabulation of numerical facts as thebasis for explanation, description and comparison of phenomena.” –Prof. Lovit

5. “Statistics is the science which deals with the methods of collecting, classifying, presenting,comparing and interpreting numerical data collected to throw some light on any sphere ofenquiry.” –SeligmanThis definition is short, simple yet comprehensive.

6. “Statistics or statistical methods may be defined as the collection, presentation, analysis andinterpretation of numerical data.” –Croxton and Cowden

The last two definitions are quite exhaustive and appear to be comprehensive. In the light of abovedefinitions of statistics, the statistical method has the following stages:

(i) Collection of data (ii) Organisation (classification) of data(iii) Presentation of data (iv) Analysis of data(v) Interpretation of data.(i) Collection of data is the first step in a statistical investigation and forms the backbone of the

investigation (or inquiry). Data should be collected for a specific and well-defined purpose verycarefully, because faulty data will lead to fallacious conclusions.

(ii) Organisation of data is done in three parts: (a) Editing, (b) Classification and (c) Tabulation.To draw meaningful conclusions, the proper organisation of the data is essential. Observed data,without proper organisation, will not lead to any conclusion.

(iii) Presentation of data. Orderly presentation of data facilitates statistical analysis. Data presented inthe form of graph and diagrams is easily understandable and has greater memorising effect andhence facilitates efficient comparison.

(iv) Analysis of data. The next step after collection, organisation and presentation of data is analysisof data. The main objective of analysis is to prepare data in such a fashion so as to arrive atcertain definite conclusions. The methods most commonly used are: Measures of Central Tendency(called measures of the first order); Measures of Dispersion (called measures of the secondorder); Skewness, Correlation, Regression, Interpolation, etc. (called measures of the third order).The analysis of facts based on observation is termed as (a) Scientific Analysis, (b) NumericalAnalysis and (c) Empirical Analysis.

(v) Interpretation of data. Interpretation or drawing valid conclusions is the last stage in a statisticalinvestigation. The interpretation of analysed data is a difficult task and needs a high degree ofskill and experience. If the analysed data are not interpreted carefully, the basic object of enquiryis liable to be defeated.


Therefore, we may give the following complete definition of statistics.Statistics (as used in the sense of data, i.e., as a plural noun) are numerical statements of facts

capable of analysis and interpretation and the science of statistics (i.e., as singular noun) is a study of theprinciples and methods used in the collection, organisation, presentation, analysis and interpretation ofnumerical data in any sphere of enquiry.

1.5 EMPIRICAL AND QUANTITATIVE ANALYSISEmpirical refers to something based on experiment, observation* or experience rather than on theory.

Analysis refers to detailed examination of statements or observations mainly to draw some conclusions.Therefore, empirical analysis can be defined as a method of studying a subject whereby knowledge isacquired as a result of actual experience.

Quantitative analysis is an attempt to lend ‘precision’ or ‘preciseness’ to the facts, so that they canbe easily compared. In quantitative analysis, numerical data are classified, tabulated and analysed to drawreasonable conclusions. As stated earlier, statistics mainly deals with the analysis of quantitative expressionsor data.

In natural sciences like physics, chemistry and biology, method of empirical analysis is commonlyused and relevant laws and statements can be tested and experimented in laboratories. But economics is asocial science wherein human behaviour is the subject of study and it covers the whole world. Empiricalanalysis is not helpful for the study of economics and other social sciences. The problems of social sciencescan be effectively studied through the technique of quantitative analysis. Under quantitative analysistechnique, the problem can be studied systematically, but the results so obtained cannot be tested in alaboratory and hence they are not so exact and accurate as the results of empirical analysis. On the otherhand, the technique of statistical analysis can be applied only when sufficient data concerning the problemunder study is available so that proper classification and tabulation for analysis can be done.

Compared to empirical analysis, statistical (quantitative) analysis has much wider applicability, almostin every field of knowledge, maybe medical science, physical science, natural science, or any other science.As far as the field of economics is concerned, the technique of statistical analysis has become so importantthat even for the understanding of elementary economics, the knowledge of statistics is essential. Marshallobserves “statistics are straws out of which I like every other economist have to make bricks.”

C.E. Engeberg remarked, “no economist would attempt to arrive at a conclusion concerning productionor distribution of wealth without an exhaustive study of statistics.”

1.6 NEED FOR QUANTITATIVE ANALYSIS (OR IMPORTANCE OF QUANTITATIVEANALYSIS)

1. Statistical analysis is useful in solving complex problems of modern business and industry.2. It provides systematic and powerful tools to decision makers to take efficient decisions based on

quantitative data.3. It enables proper allocation of resources.4. It helps in minimising waiting and servicing cost.5. It enables the management to decide when to buy and how to buy.6. It helps in choosing an optimum strategy, i.e., the action which can give maximum benefit at

mmimum cost.

* Any recording of information whether it is numerical or categorical, is observation.


7. It renders great help in the optimum allocation of resources.8. Management can know the reactions of the integrated business system through quantitative

analysis.

1.7 SCOPE OF STATISTICSScope of statistics is studied under the following three heads:

1. Nature of Statistics2. Subject-matter of Statistics3. Limitations of Statistics.

1. Nature of StatisticsBy nature of statistics, we are to determine whether statistics is an art or a science. Statistics is

regarded as an art of applying the science of scientific methods. In statistics, we not only study differentmethods of studying a problem but also study how those methods should be applied in different situations.

In the words of Tippett, “Statistics is both a science and an art. It is a science in that its methods arebasically systematic and have general application; and an art in that their successful application depends,to a considerable degree, on the skill and special experience of the statistician, and on his knowledge ofthe field of application.”

2. Subject-matter of StatisticsSubject-matter of statistics is divided into two major parts:(i) Descriptive statistics

(ii) Inferential or inductive statistics.(i) Descriptive statistics. The descriptive statistics consists of those statistical methods which tell us

how to describe the characteristics of a body of data. They deal with the collection, tabulation and presentationof data and the calculations of measures which describe the data in various ways. Descriptive statistics canbe either univariate or bivariate. Following are included in the univariate descriptive statistics:

1. Frequency distributions2. Measures of central tendency3. Measures of variability, i.e., dispersion.

The study of correlation and regression are a part of bivariate descriptive statistics.(ii) Inferential or inductive statistics. Inferential statistics deals with those methods where conclusions

about a large group are drawn by studying a part of it. For example, by checking a few grains of rice, ahousewife makes an estimate whether the whole lot has been cooked.

3. Limitations of StatisticsStatistics is an extremely useful science, but it has its limitations and shortcomings too. Newshome

writes, “It (statistics) must be regarded as an instrument of research of great value, but having severelimitations, which are not possible to overcome and as such, they need our careful attention.” Tippetobserves, “The application of statistical methods to investigation in the technological and indeed in anyother field is based on assumptions, is subject to limitations and often leads to uncertain results.” Theselimitations should be kept in mind while using or interpreting statistics. These limitations are:


(1) Statistics studies only the quantitative aspect of a problem and does not study its qualitativeaspects. The science of statistics studies only the quantitative aspect of a problem. The factswhich are not capable of being quantitatively measured, like, intelligence, poverty, honesty, etc.cannot be studied unless these attributes are reduced to precise quantitative terms as Prof. Secristobserves, “Some phenomena cannot be quantitatively measured, honesty of resourcefulness,integrity, goodwill, all important in industry as well as in life generally, are not susceptible ofdirect statistical measurement.” Prof. Bowley also states that, “They (statisticians) can at bestmeasure only the numerical aspect of a phenomenon while very often they must be contendedwith measuring not the facts they wish but some allied quantity.” This limitation of statisticslimits the scope of the science.

(2) Statistics deals with averages. According to W.I. King, “Statistics largely deals with averagesand these averages may be made up of individual items radically different from each other.”Laws of statistics are true on an average. W.I. King states, “no system which makes a large orcomplex group intelligible to the mind at a glance can avoid effacing most of the minorirregularities.”

(3) Statistics does not study individuals. According to Tippett, “Statistics is essentially totalisation,because it is not concerned with individual values–but only with classes.” Statistics does notstudy individual items. It deals with mass phenomena. This is a serious limitation of statistics. Inthe words of W.I. King, “Statistics from their very nature of subject cannot and will never beable to take into account individual cases. When these are important, other means must be usedfor their study.”

(4) Statistical results are only approximately correct. Statistics unlike mathematics or accountancy,is a science of estimates. According to Dr. Bowiey, “When observations are extended, manysources of inaccuracy are found to be present, and it is very frequently impossible to removethem completely. Statistical results are, therefore, very generally estimates rather than exactstatements.” Statistics fails in the fields where perfect accuracy is desired.

(5) Statistical results are not always beyond doubt. According to Prof. Connor, “Statistics deal onlywith measurable aspects of things and therefore can seldom give the complete solution to aproblem. They provide a basis for judgement but not the whole judgment.” Dr. Bowley has alsoexpressed the same views. According to him, “Statistical results are essential, when judgment isto be formed on any question that involves numbers, quantities or values, but they must alwaysbe brought into relation with personal, political, aesthetic or other non-quantitativeconsiderations that may be of greater importance in deciding on a course of action.” Too muchdependence on statistics may be misleading.

(6) Statistics is only a means and not an end. It analyses the facts and throws light on the realsituations. D. Gregory and H. Ward observe, “Statistics cannot run a business or government.Nor can the study of statistics do more than provide a few suggestions or offer a few pointers as

Limitations of Statistics

Deals withquantitativecharacteris-

tics only

Dealswithaver-ages

Doesnot

studyindi-

viduals

Resultsare

approxi-matelycorrect

Resultsnot

alwaysbeyonddoubt

Only ameans

and notan end

Misusepossible

Shouldbe

usedonly byexperts

Not theonly

method

Data shouldbe homoge-neous and

uniform


to firm’s or a government’s future behaviour.” Statistical technique is no doubt essential forprocessing, analysing and interpreting numerical data. But the methods are by no means perfectand it is very dangerous to jump to conclusions without a clear-cut idea about its limitations.

(7) Misuse of statistics is possible. W.I. King says, “One of the shortcomings of statistics is thatthey do not bear on their face the label of their quality. Statistics is like clay and they can bemoulded in any manner. Statistics can be misused by deliberately twisting or manipulatingdata. That is why some people say ‘statistics can prove any thing’ or ‘statistics is another formof lying’.”

(8) Statistics should be used only by experts. Yule and Kendall point out, “statistical methods aremost dangerous tools in the hands of an inexpert.” Only well trained people having goodknowledge of statistics should use statistics, otherwise the results are liable to be misleadingand dangerous.

(9) Statistical study is not the only method. According to Croxton and Cowden, “it must not beassumed that the statistical method is the only method to use in research, neither should thismethod be considered the best attack for every problem.” Results obtained by statistical methodsshould be verified by other methods.

(10) Homogeneity of data. For comparison purpose, data should be homogeneous and uniform.Quantities measured in different units cannot be compared. Since foodgrains and cloths are measuredin different units, we can simply compare their value in rupees but not their quantities.

1.8 FUNCTIONS OF STATISTICSThe functions of the science of Statistics are beautifully summed up by Robert W. Burgess in the

following words:“The fundamental gospel of statistics is to push back the domain of ignorance, rule of thumb,

arbitrary or premature decisions, tradition and dogmatism and to increase the domain in which decisionsare made and principles are formulated on the basis of analytical quantitative facts.”

If we analyse the above statement, we would find that the main functions of the science of statisticsare as given below:

1. To simplify complex facts.2. To provide comparative study.3. To study relationship between different facts.4. To enlarge individual knowledge and experience.5. To formulate policies in different fields.6. To measure the effects.7. To test a hypothesis.8. To provide numerical measurements.9. To forecast (to indicate trend behaviour).

10. To classify data.11. To measure uncertainty.12. To draw valid inferences.(1) To simplify complex facts. The raw data is often unwieldy and complex. The function of statistical

methods is to simplify large mass of numerical data and present them in a more intelligible form.


Dr. Bowley states that, “the object of a statistical estimate of complex groups is to present anoutline to enable the mind to comprehend with a single effort the significance of the whole.” Inthe words of W.I. King, “It is for the purpose of simplifying these unwieldy masses of facts thatstatistical science is useful. It reduces them to numerical totals or averages which may beabstractly handled like any other mere numbers. It draws pictures and diagrams to illustrategeneral tendencies and thus in many ways adapt these groups of ideas to the capacity of ourintellects.”

(2) To provide comparative study. The science does not merely mean counting but also comparison.Statistical devices like averages, ratios, percentages, rates, coefficients, standard error, etc. offerthe best way of comparison between two phenomena. Boddington states that, “The object ofstatistics is to enable comparison to be made between past and present results with a view toascertaining the reasons for changes which have taken place and the effect of such changes inthe future.” Dr. Bowley observes, “A chief practical use of statistics is to show relativeimportance, the very thing which an individual is likely to misjudge. Statistics are almostalways comparative.”

(3) To study relationship between different facts. Another function performed by statistics is toinvestigate relationship between two or more phenomena. The relationship existing between demandand supply, rainfall and agricultural production can best be measured with the help of statisticalmethods.

(4) To enlarge individual knowledge and experience. The science of statistics enlarges humanknowledge and experience by making it easier for man to understand, describe and measure theeffects of his action on the action of others. Statistical methods have provided such a master keyto mankind that we can use them anywhere and can study any problem in its correct perspectiveand on right lines.In the words of Whipple, “Statistics enables one to enlarge his horizon.” Dr. Bowley alsoobserves that, “The proper function of statistics, indeed, is to enlarge individual experience.”Bowley further observes, “a statistical estimate may be good or bad, accurate or reverse, but inalmost all cases it is likely to be more accurate than a casual observer’s impression, and in thenature of things, can only be disproved by statistical methods.”Precise understanding of the concepts like national income, population, agricultural production,industrial production, etc. can be possible by using statistical technique.

(5) To formulate policies in different fields. Statistics helps in formulating policies in different fields,specially in social, economic and business fields. Malthus’ Theory of Population and Engels’Law of Family Expenditure are due to the statistics collected by these two noted economists.The governmental policies are framed on the basis of statistics. Policies in the fields of taxation,import-export, social welfare, wage, etc. are formulated on the basis of statistical data.

(6) To measure the effects. Not only in formulating a policy but also in measuring the effects of apolicy, statistics act as guide. The effects of a change in the Bank Rate or tax rate, etc. can bethoroughly studied with the help of statistics.

(7) To test a hypothesis. The most important theoretical function of statistics is to test the hypothesisin theory and discover a new theory. According to Prof. J.M. Keynes, “The function of statisticsis first to suggest empirical laws which may or may not be capable of subsequent deductiveexplanation and secondly to supplement deductive reasoning by checking its results andsubmitting them to the best of experience. The correctness of the laws of the different branchesof knowledge can only be tested with the help of statistical data and statistical techniques.


(8) To provide numerical measurements. Facts expressed in numbers are easily understood. It is oneof the major functions of statistics to express the facts relating to different phenomena in numbers.For example, if we say that agricultural production of India is increasing, it does not carry muchmeaning but if we give year-wise figures, it becomes easier even for a common man to understandindex numbers also. In the last few decades, it has been made possible to express even thequalitative phenomena in the numerical forms to give them more concrete shape.

(9) To forecast. Making estimates for future on the basis of past and present behaviour is calledforecasting. Extrapolation is a statistical method which helps in forecasting. Forecasting can alsobe done with the help of trend equations. We can forecast food production, growth of population,supply of power, etc. on the basis of past records.

(10) To classify data. Statistical methods help us to arrange the data in groups according to theircommon characteristics. This facilitates analysis and comparisons.

(11) To measure uncertainty. Statistics helps us to take decisions in the face of uncertainty and studythe chance of occurrence of certain events. In theory of games, we take decision in the face ofuncertainty. John I. Griffin observes, “Statistical knowledge is, however, imperfect and alwayscharacterised by uncertainty. Such imperfect and uncertain knowledge is not to be rejected asworthless because statistical methods permit the measurement of the imperfection anduncertainty.”

(12) To draw valid inferences. In the fields of Business and Economics, many times rational and validconclusions are drawn by collecting and analysing the available facts.

1.9 USES AND IMPORTANCE OF STATISTICSThe words of H.G. Wells, “Statistical thinking will one day be as necessary for efficient citizenship

as the ability to read and write” are proving true in present-day life. Today, there is hardly any sphere oflife which is not affected by statistical methods in one way or the other. In fact, statistics has become an“arithmetic of human welfare”. Commenting on the importance of statistics, Croxton and Cowden write, “Themethods of statistics are useful in ever widening range of human activities in any field of thought in whichnumerical data may be had... . Today, there is hardly a phase of endeavour which does not find statisticaldevices at least occasionally useful.” Tippet says, “Statistics affects every body and touches life at manypoints.” Though statistics deals with quantitative data but qualitative data, for example, attributes such asdefectives, standard, deluxe, new and used can be transformed into number of defectives, proportionals ofstandard models and so forth. In this manner, extensive statistical analysis can be made of qualitative as wellas quantitative data.

1.10 RELATIONSHIP OF STATISTICS WITH OTHER SCIENCES

Importance of Statistics in EconomicsStatistical data are a powerful aid in economic analysis. The study of every economic problem requires

the use of statistical methods just as a doctor needs his stethoscope to examine almost each and everypatient.

In the words of Ya-Lun-Chou, “Economists depend upon statistics to measure economic aggregatessuch as gross national output, consumption, saving, investment, expenditure and changes in the value ofmoney. They also use statistical method to verify economic theory and to test hypothesis.” Most of theeconomic problems can be expressed numerically which helps the economist in collecting data and applyingstatistical methods in formulating policies and arrive at certain decisions. In fact, statistical data and statisticalmethods are the tools and appliances of an economist’s laboratory. This led Prof. Marshall, to say, “Statisticsare the straws out of which I like every other economist have to make bricks.”


Statistics of consumption enable us to find out the ways in which people of different classes ofsociety spend their incomes. Such statistics are very helpful for knowing the standard of living and thetaxable capacity of the people.

In distribution too, statistics play a vital role. With the help of statistics, the national wealth of thecountry is estimated and its distribution among the people of the country is found out. Equitable distributionof national income and wealth is another serious problem to be best studied statistically.

In the field of exchange we study markets, laws of prices based on supply and demand, cost ofproduction, etc. A systematic study of all these facts cannot be made without statistics. What price shoulda monopolist charge in order to reap the maximum profit? What shall be the price of a particular commodityif its supply is increased or decreased?–All such questions can best be answered with the help of statistics.Statistics are the foundation stone of the theory of exchange.

Statistics of production help us to adjust the supply according to demand. Such statistics are themeasures of productivity of a country. Thus, productivity of different factors of production can be gaugedand compared. Statistics of production of an area or a country can be compared with another area or anothercountry. Census of production helps in understanding the economy and taking corrective measures wherevernecessary.

Public Finance. Public finance studies the revenue and expenditure activities of the State. The statementwhich shows the revenue and expenditure of the State is known as ‘Budget’. Budget is a statistical document.Fiscal policy, deficit financing, nature and amount of subsidies, etc. are concepts which are based onstatistics.

Monetary Policy. Monetary policy is the policy of central bank of a country. Bank rate, selective creditcontrols, open market operations, cash reserve ratio, etc. are the chief weapons of monetary policy. All theseconcepts are statistical in nature.

Macroeconomic Concepts. The worldwide recession of 1930s necessitated the study of nationaleconomic aggregates. What should be the increase in national income if the investment in the country isincreased by ` 100 crores? How much consumption will be affected if the national income increases ordecreases by some fixed amount?–All these concepts are statistical in nature. The modern economist lays agreat stress on all these concepts and theories.

Econometrics. Econometrics is a product of the 20th century. It is the combination of economics,mathematics and statistics. It is assumed that with the help of econometrics, economics will become an exactscience.

Input-output Analysis. Input-output analysis is based on statistical data, which explain the relationshipbetween the input and the output. Modern theory of probability, sampling, correlation and regression, timeseries, index numbers, etc. are the concepts used in economic analysis.

The Law of Demand and the Elasticity of Demand are based on inductive reasoning. Under inductivereasoning, general laws are propounded by studying data about a particular phenomenon. It is the functionof a statistician to collect data about a particular phenomenon. From the data collected by statisticians, lawsare derived by economists.

Modern economists are trying to make economics as an exact science. In this effort, use of statisticsand statistical methods is inevitable. W.A. Spur and C.P. Bonini observe, “Economists today are no longercontent to theorize in abstract term, citing statistics only as needed to buttress their arguments. Instead,they utilise the excellent data now available to build a sound factual foundation for their reasoning.”

A statistical approach to an economic problem not only leads to its correct description but alsoindicates lines along which it is to be tackled. Hence, in all types of economic problems, statistical approach


is essential and statistical analysis is useful. To quote Dr. Bowley, “No student of Political Economy canpretend to know complete equipment unless he is master of the methods of statistics, knows its difficulties,can see where accurate figures are possible, can criticise the statistical evidence and has an almostinstinctive perception of the reliance that he may place on the estimates given to him.” And C.E. Engebergsays, “No economist would attempt to arrive at a conclusion concerning the production or distribution ofwealth without an exhaustive study of statistical data.” Justice Oliver Vendell Holmes remarked that, “Theblack letter man may be the man of the present, but the man of the future is the man of Statistics and themaster of Economics.”

Thus, we find that there is kinship between Economics and Statistics.

Importance of Statistics in Planning“Economic planning means making of major economic decisions–what and how much is to be

produced, and to whom it is to be allocated–by the conscious decisions of a determinate authority on thebasis of a comprehensive survey of the economy as a whole.” It implies a conscious and deliberate action toattain certain predetermined economic aims. In planning, statistics plays an indispensable role. In order tounderline the role of statistics in economic planning, we can do no better than to quote Baykon, a Russianeconomist, that “Every act of planning presupposes a preliminary investigation of existing conditions.”

Statistics are used in economic planning for the following purposes:1. To know the stage of economic development in our country as compared to other countries,

statistics are of great help. The problems like overpopulation, unemployment, lack ofindustrialisation, low rate of capital formation, etc. which are the characteristics of undevelopedand underdeveloped countries, can be fully understood only with the help of statistical facts.

2. Statistical language is essential to understand the effect of determining factors of economicdevelopment in the past–what psychological and sociological factors need to be developed foreconomic progress, how far we have progressed in the technical field, what is the level ofproductivity and such other questions.

3. It is necessary to determine priorities in different sectors for development. This can be done byexamining the structure and importance of production, consumption, exchange and distributionsectors of the economy. For determining priorities, statistical methodology is essential.

4. Economic planning is done to achieve predetermined objectives and targets. They have to beexpressed in quantitative terms. Financial planning is a part of economic planning. Financialresources are mobilised through taxation, public saving, deficit financing, debt issues, etc. Planningand budgeting are integrated. All these functions can be performed successfully with the help ofstatistics.

5. Evaluation of the success of a plan is also done with statistics.Thus, we see planning without proper and adequate statistics is unconceivable. “planning without

statistics is like a ship without rudder and a compass.”Tippet says, “Planning is the order of the day and without statistics, planning is inconceivable.”

Importance of Statistics in BusinessBusiness is regarded as a profession of forecasting which requires a high degree of skill. Success in

business depends upon precision in forecasting. A businessman must make a proper and scientific analysisof the past records in order to forecast the future business conditions. Business barometers are constructedwith the help of existing records to tell us the future course of business events.


Every businessman, irrespective of his nature of business, has to employ statistical techniques inestimating trend of prices, trend of economic activities, etc.

A producer estimates probable demand of his goods, analyses the effects of trade cycles and seasonalvariations as also of changes in habits and customs of people on the demand of his wares, and after takingall these factors into consideration finally takes decision about the quantum of production. Buddingtonsays, “In order to succeed in any business today, the businessman must study all the factors which enterinto production, buying and selling, exporting and importing of goods in which he deals.”

Buddington further says, “A successful businessman is one whose estimates most closely approachaccuracy.” Accuracy can be achieved only by proper analysis of past records with the help of statisticaltools. Even the industrialists, farmers, and manufacturers take decisions about the volume of future productionon the basis of past records and trends. Again, we may quote Ya-Lun-Chou, “It is not an exaggeration tosay that today nearly every decision in business is made with the aid of statistical data and statisticalmethods.” We may conclude that industry and business involve risk and when it is risk, it is better to havea calculated risk and the risk can be calculated with the help of statistical-tools.

Importance of Statistics in State and AdministrationImportance of statistics in State and administration started its origin as ‘science of Kings’ or ‘political

arithmetic’. But now, in fact, statistics has become the eyes of the Government administration.W.M. Harper observes, “Increasingly, figures have become the basis of rational decisions, and

events are proving that these decisions based on figures give better results.” Finance Minister (i) preparesbudget by using the data relating to revenue and expenditure, (ii) takes decisions for considering new taxesand planning public expenditure on the basis of the data concerning production, import, export, nationalincome, etc.

It is the duty of the Government to promote economic development, reduce inequalities in thedistribution of wealth and income and maintain stability and full employment. The Government decidesmeasures to achieve these objectives on the basis of statistical data.

Not only in peace, even during war and periods of crises, natural calamities, etc. statistical records arehelpful in framing suitable policies.

Government formulates its policies concerning family planning, nationalisation, establishment of newindustries, etc. on the basis of statistical data.

Reports of different commissions and committees appointed by the Government are always substantiatedby the statistical facts.

Government makes assessment about the efficiency of different departments like health, education,industry, agriculture, etc. on the basis of statistical data.

1.11 DISTRUST OF STATISTICSDistrust means lack of faith and confidence. Distrust in statistics means lack of faith and confidence in

statistical statements and methods. About statistics people often quote derogatory remarks like–(i) Statistics are lies of the first order.

(ii) An ounce of truth will produce tonnes of statistics.(iii) Statistics can prove or disprove anything.(iv) Statistics can prove nothing.(v) Figures do not lie.

(vi) If figures say so, it cannot be otherwise.


(vii) Statistics is another form of lying.(viii) There are three kinds of lies–lies, damned lies and statistics.

(ix) Statistics are like bikinis. They reveal what is interesting and conceal what is vital.It is only the misuse of statistics which results in its distrust. Prof. Ya-Lun-Chou explains it as,

“Derogatory statements have been inspired by the misuse of statistics on the part of untrained practitionersof statistical methods, ignorant users of statistical results, dishonest people who employ statistics tomislead and confuse rather than to clarify and illuminate.” King rightly highlights, “Science of statistics isthe most useful servant but only of great value to those who understand its proper use.” For example,medicines are used for curing diseases, but wrong administration of a medicine may prove poisonous andfatal. For such unpleasant results, medicine cannot be blamed. Similarly, if the statistical facts are misinterpretedby the people to serve their ends, we may be dishonest to blame the science of statistics as such. Thus,statistics are like clay of which one can make a God or Devil as one wishes. The problem of distrust ofstatistics arises because of its misuse. Jule and Kandall have rightly said, “Statistical methods are mostdangerous tools in the hands of an inexpert.” Statistics should be used for a useful purpose and not as anexcuse for making wrong interpretations.

According to F.C. Mills, “Statistics, as a tool, requires proper usage and the results obtained throughstatistical analysis require intelligent interpretation.”

From the above words of Mills, it is clear that Statistics is useful if used properly. Sir F. Galton has alsosaid, “Some people hate the very name of Statistics but I find them full of beauty and interest. Wheneverthey are not brutalized but delicately handled by the higher methods and are properly interpreted, theirpower in dealing with complicated phenomenon is extraordinary.” Distrust of statistics can be removed ifit is used judiciously and by expert persons having no prejudice and bias. Prof. J.M. Keynes writes, “If thelimitations of statistics are clearly recognised, if they are accurately collected over an adequate range, ifthey are employed without prejudice and after full enquiry into their true significance, and if they arefairly and properly grouped, then their value is unique and the statistical method easily makes good itsclaim to rank as a most effective and reliable instrument of science.”

QUESTIONS

Very Short Answer Type Questions

1. Explain three features of statistical data2. Explain 2 more serious limitations of statistics3. Explain the importance of statistics in business4. Explain three features of statistics

A. Choose the correct option:

1. Statistics as a singular noun means _________.(a) statistical data (b) statistical methods(c) inductive statistics (d) descriptive statistics

2. Statistics as a plural noun indicates _________.(a) statistical methods (b) a bundle of lies(c) statistical data (d) Fool’s Paradise

3. Distrust of statistics is due to _________.(a) misuse of statistics (b) insufficient statistical methods(c) limited scope of statistics (d) people’s impression that it is useless


4. Statistics facilitates _________.(a) comparison of data (b) collection of data(c) organisation of data (d) disposal of data

5. Statistics is the science of analysing _________.(a) qualitative data (b) quantitative data(c) any kind of data (d) only very simple facts

Answers:1. (b), 2. (c), 3. (a), 4. (a), 5. (b)

B. Very Short Answer Type Questions:1. Explain statistics as a plural noun.2. Define empirical analysis.3. What is inductive statistics?4. Define descriptive statistics.5. What is Direct Personal Investigation? Give its merits and demerits.6. Define and explain Primary Data with examples.7. Enlist the preliminary steps you would take in planning statistical investigation.8. Name the main sources of Secondary Data.9. Name the main method of collecting Primary Data.

10. Write short note on points to be kept in mind while drafting a questionnaire.11. What are the units in collection of data?

C. Short Answer Type Questions:

1. Mention four important functions of statistics.2. Explain statistics in singular and plural sense.3. What are the two meanings of statistics?4. Define statistics.5. Explain the importance of statistics for the Government.6. What does statistical thinking deal with?7. Write a note on distrust of statistics.8. What is the importance of statistics in business?9. What is the importance of statistics in planning?

10. What is the importance of statistics in Economics?11. “Statistics affects everybody and touches life at many places.” Explain.12. What is statistics of consumption and statistics of production?13. Explain Inferential statistics.14. Write four limitations of statistics.15. Write a short note on scope of statistics.16. “Statistics is regarded as a science of counting.” Explain.17. “No isolated facts constitute statistics.” Explain.18. “Statistics is the science of estimates and probabilities.” Explain.19. Why statistics is regarded as both science and art?


20. Why statistics is not trusted in present business world?21. Briefly indicate the characteristics of statistics.22. State the types of statistical units.23. State the features of a good statistical unit.24. Distinguish between statistical method and experimental method.25. State any three situations of distrust of statistics.26. State the need of statistical analysis.27. How statistics is helpful for natural science?28. How will you plan for a statistical survey?

D. Long Answer Type Questions:1. Define statistics. State the uses of statistics in business.2. Explain the limitations of statistics.3. Explain the functions of statistics.4. Explain distrust of statistics and how it can be removed.5. Illustrate the need for empirical analysis in economics with two examples.6. “Statistical analysis is of vital importance for successful businessmen, economists, administrators

and educationists.” Discuss with illustrations.7. “Statistical thinking is as necessary for efficient citizenship as the ability to read and write.”

Explain this statement in about 200 words.8. “A well-wrapped statistics is better than Hitler’s ‘Big-lie’. It misleads, yet it cannot be pinned on

you.” Discuss.9. Discuss the importance of statistics.

E. State whether the following statements are True or False:1. Statistics is concerned with quantitative data only.2. Quantitative data lends preciseness to the facts.3. As a singular noun, statistics means statistical data.4. Any single numerical figure is statistics.5. All statistics are numerical statements.6. Statistics can be misused.7. Statistics do not ignore the quantitative aspects.8. Statistics deals with collection of data and not its presentation.9. Statistics helps in forecasting.

10. Statistical analysis is applicable in all investigations wherefrom quantitative data can be obtained.11. Statistics is the science of analysing quantitative data.12. Statistics are numerical statements of facts, but all facts numerically stated are not statistics.13. Statistics is both singular and plural.14. Statistics affects everybody and touches life at many points.15. Statistics is always concerned with mass phenomena and never with a singular observation.

Answers:1. (T), 2. (T), 3. (F), 4. (F), 5. (T), 6. (T), 7. (F), 8. (T), 9. (T), 10. (T), 11. (T), 12. (T), 13. (T) 14. (T),15. (T)


F. Fill in the Blanks:

1. Statistics as a _________ noun means the same thing as data. (singular, plural)2. Statistics are _________ statements. (qualitative, quantitative)3. _________ is the first step in a statistical investigation. (Collection of data, Organisation of data)4. The nature of statistical inference is _________. (deductive, inductive)5. The Statistics as a _________ noun refers to statistical methods. (singular, plural)6. _________ includes condensation and summarisation of data. (Analysis, Interpretation)7. _________ facilitates comparison of data. (Economics, Statistics)

Answers:1. plural, 2. quantitative, 3. Collection of data, 4. inductive, 5. singular, 6. Analysis, 7. Statistics.

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